Question: Simplify and expand the following expression: $ \dfrac{1}{a - 9}+\dfrac{3a - 9}{a + 7} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(a - 9)(a + 7)$ Multiply the first term by $\dfrac{a + 7}{a + 7}$ $ \begin{align*} \dfrac{1}{a - 9} \times \dfrac{a + 7}{a + 7} & = \dfrac{(1)(a + 7)}{(a - 9)(a + 7)} \\ & = \dfrac{a + 7}{(a - 9)(a + 7)}\end{align*} $ Multiply the second term by $\dfrac{a - 9}{a - 9}$ $ \begin{align*} \dfrac{3a - 9}{a + 7} \times \dfrac{a - 9}{a - 9} & = \dfrac{(3a - 9)(a - 9)}{(a + 7)(a - 9)} \\ & = \dfrac{3a^2 - 36a + 81}{(a + 7)(a - 9)}\end{align*} $ Now we have: $ = \dfrac{a + 7}{(a - 9)(a + 7)} + \dfrac{3a^2 - 36a + 81}{(a + 7)(a - 9)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{a + 7 + 3a^2 - 36a + 81}{(a - 9)(a + 7)} $ $ = \dfrac{-35a + 88 + 3a^2}{(a - 9)(a + 7)}$ Expand the denominator: $ = \dfrac{-35a + 88 + 3a^2}{a^2 - 2a - 63}$